196 CHAPTER VII. 



61 % 62 



5~D f ^ O O \ f A" 



Jtl %(()(}& (JpCC) K\6yO 



P Q x({la ]}a) x(y~d 



- Gay a'a - y'y 



where ~ 1 ^0 r c? and 



Since every coefficient of TF equals its own conjugate with respect 

 to the 6rF[p w ], T7 belongs to that field. 



As in 196, the substitutions U form a group {7} holoedrically 

 isomorphic with the group of binary linear substitutions of determinant 

 unity in the GF\_p* n ~\. The substitutions W form an isomorphic 

 group {W} leaving f absolutely invariant and therefore a subgroup 

 of E^ p n. Indeed, if we take Z7~TFand U'~W, then to UU' will 

 correspond 



since the set of substitutions U is commutative with the set U by 

 196. Moreover, an identity UU^U'U' or U'- 1 U=U'U~ 

 requires U 1 = U or CU f where C merely changes the signs of the 

 four indices. In fact, the groups {U} and {U} have in common only 

 the identity and C. Hence CU is the only substitution in addition 

 to U which corresponds to the product W ~ U-JJi= C U^ - CU V It 

 follows that the quotient - group of {U} by {/, C} is holoedrically 

 isomorphic both with the simple linear fractional group LF(2,p'* n ') 

 and with the group {W\ In particular, the order of {W} is 

 JL(jp4_l)p2n or (2 4n l)2 2w according as p > 2 or p = 2. For 



>> 2, p n > 3, IJ^n has the order (p* n + p n ) (p 2n l}p n , being holo- 

 edrically isomorphic with O v (4,^) n ), whose order is given in 172. 

 For p = 2, jEJ 4) g* is holoedrically isomorphic with the group leaving 

 i%+ 2% + ^i? + ^i absolutely invariant, whose order is shown in 

 204 to be 2(2 4 -l)2 2w . Hence {W} is of index 2 under E^n. 

 According as p > 2 or p = 2, { W } is extended to E^ p n by T 2 , x 

 or (li%), where ^ is any not -square in the GrF[p n ]. It is only 

 necessary to show that these substitutions are not of the form W, 

 If (gj%) were of the form TF, then <yy = p]} = 0, S = Q = 0. Hence 

 /3 = y = 0, ^"ad = <ydo", ^ad = (?a^. Hence would c?da" and 



